GLMM example 5
Murray Logan
16 October, 2021
Preparations
Load the necessary libraries
library(car) # for regression diagnostics
library(broom) # for tidy output
library(ggfortify) # for model diagnostics
library(sjPlot) # for outputs
library(knitr) # for kable
library(effects) # for partial effects plots
library(emmeans) # for estimating marginal means
library(MASS) # for glm.nb
library(MuMIn) # for AICc
library(tidyverse) # for data wrangling
library(brms)
library(tidybayes)
library(bayesplot)
library(broom.mixed)
library(rstan)
library(patchwork)
library(DHARMa)
library(standist) # for visualizing distributions
library(rstanarm)
library(ggeffects)
library(DHARMa)Scenario
Some ornithologists were interested in the degree of sibling negotiations in owl chicks. Specifically, they wanted to explore how sibling negotiations were affected by feeding satiety and the sex of the parent returning to the nest. The ornithologists had accessed to a number of owl nests and were able to count (via recording equipment) the number of sibling negotiations (calls) that the owl chicks made when the parent returned to the nest.
We could hypothesise that the chicks might call more if they were hungry. As part of the investigation, the researchers were able to provided supplementary food. As such, they were able to manipulate the conditions such that sometimes the chicks in a nest would be considered deprived of supplementary food and at other times they were satiated.
As a parent returned, the researchers recorded the number of sibling negotiations (calls) along with the sex of the parent. Since the number of calls is likely to be a function of the number of chicks (the more chicks the more calls), the researchers also counted the number of siblings in the brood.
Each nest was measured on multiple occasions. Hence, we must include the nest as a random effect to account for the lack of independence between observations on the same set of siblings.
Read in the data
owls <- read_csv("../public/data/owls.csv", trim_ws = TRUE)
owls %>% glimpse()## Rows: 599
## Columns: 7
## $ Nest <chr> "AutavauxTV", "AutavauxTV", "AutavauxTV", "Autavaux…
## $ FoodTreatment <chr> "Deprived", "Satiated", "Deprived", "Deprived", "De…
## $ SexParent <chr> "Male", "Male", "Male", "Male", "Male", "Male", "Ma…
## $ ArrivalTime <dbl> 22.25, 22.38, 22.53, 22.56, 22.61, 22.65, 22.76, 22…
## $ SiblingNegotiation <dbl> 4, 0, 2, 2, 2, 2, 18, 4, 18, 0, 0, 3, 0, 3, 3, 6, 0…
## $ BroodSize <dbl> 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, …
## $ NegPerChick <dbl> 0.8, 0.0, 0.4, 0.4, 0.4, 0.4, 3.6, 0.8, 3.6, 0.0, 0…Data preparation
Let start by declaring the categorical variables and random effect as factors.
## Amount of Sibling negotiation (vocalizations when parents are absent)
## Foot treatment (deprived or satiated
## Sex of parent
## Arrival time of parent
## Nest as random
## Brood size offset
owls <- owls %>% mutate(
Nest = factor(Nest),
FoodTreatment = factor(FoodTreatment),
SexParent = factor(SexParent),
NCalls = SiblingNegotiation
)Exploratory data analysis
As the response represents counts (the number of calls), it would make sense to start by considering a Poisson model. We could attempt to model the response as the number of calls divided by the brood size, but this would result in a response that has no natural distribution.
Instead, if we include brood size as an offset, it will standardise the effects according to brood size (similar to having divided the response by brood size), yet retain the Poisson nature of the response.
The effects of offsets, unlike regular covariates, are not estimated. Rather they are assumed to be 1 (on the link scale). This means that since Poisson uses a log link, then the offset should be of a logged version of the brood size.
Model formula: \[ y_i \sim{} \mathcal{Pois}(\lambda_i)\\ ln(\lambda_i) =\boldsymbol{\beta} \bf{X_i} + \boldsymbol{\gamma} \bf{Z_i} \]
where \(\boldsymbol{\beta}\) and \(\boldsymbol{\gamma}\) are vectors of the fixed and random effects parameters respectively and \(\bf{X}\) is the model matrix representing the overall intercept and effects of food treatment, sex of parent, arrival time (and various interactions) on the number of sibling negotiations. Brood size was also incorporated as an offset. \(\bf{Z}\) represents a cell means model matrix for the random intercepts associated with individual nests.
Perhaps we could start off by exploring the main fixed effects. To mimic the log-link, we will use a log-transformed y axis. Since there may well be zeros (no calls detected), we will use a pseudo log scale). We will also include the raw data (jittered and dodged)
ggplot(data = owls, aes(y = NCalls, x = FoodTreatment, color = SexParent)) +
geom_violin() +
geom_point()
ggplot(data = owls, aes(y = NCalls, x = FoodTreatment, color = SexParent)) +
geom_violin() +
geom_point(position = position_jitterdodge(jitter.width = 0.2, dodge.width = 0.9))
ggplot(data = owls, aes(y = NCalls, x = FoodTreatment, color = SexParent)) +
geom_violin() +
geom_point(position = position_jitterdodge(jitter.height = 0, dodge.width = 1)) +
scale_y_continuous(trans = scales::pseudo_log_trans())
Now, a similar plot separated for each nest.
ggplot(data = owls) +
geom_point(aes(y = NCalls, x = FoodTreatment, color = SexParent), position = position_dodge(0.5)) +
facet_wrap(~Nest)
It might also be worth establishing that there is a linear relationship between the number of calls and brood size. Again, we will mimic the use of the log-link by transforming the axes.
ggplot(data = owls, aes(y = NCalls, x = BroodSize, color = SexParent)) +
geom_point() +
geom_smooth(method = "lm") +
facet_grid(~FoodTreatment) +
scale_y_continuous(trans = scales::pseudo_log_trans()) +
scale_x_log10()
Fit the model
rstanarm
Using default priors
In rstanarm, the default priors are designed to be weakly informative. They are chosen to provide moderate regularisation (to help prevent over-fitting) and help stabilise the computations.
owls.rstanP <- stan_glmer(NCalls ~ FoodTreatment * SexParent +
offset(log(BroodSize)) + (1 | Nest),
data = owls,
family = poisson(link = "log"),
refresh = 0,
iter = 5000,
warmup = 2000,
thin = 10,
chains = 3,
cores = 3
)owls.rstanP %>% prior_summary()## Priors for model '.'
## ------
## Intercept (after predictors centered)
## ~ normal(location = 0, scale = 2.5)
##
## Coefficients
## Specified prior:
## ~ normal(location = [0,0,0], scale = [2.5,2.5,2.5])
## Adjusted prior:
## ~ normal(location = [0,0,0], scale = [5.01,5.08,5.68])
##
## Covariance
## ~ decov(reg. = 1, conc. = 1, shape = 1, scale = 1)
## ------
## See help('prior_summary.stanreg') for more detailsThis tells us:
- for the intercept (when the family is Gaussian), a normal prior with a mean of 0 and a standard deviation of 2.5 is used. The 2.5 is used for all intercepts. This is then scaled to the scale of the data by multiplying by the standard deviation of the response.
2.5 * sd(owls$NCalls)## [1] 16.67704- for the coefficients (in this case, just the difference between strong and weak inoculation), the default prior is a normal prior centred around 0 with a standard deviation of 2.5. This is then adjusted for the scale of the data by multiplying the 2.5 by the ratio of the standard deviation of the response by the standard deviation of the numerical dummy variables for the predictor (then rounded).
model.matrix(~ FoodTreatment * SexParent, data = owls) %>%
apply(2, sd) %>%
(function(x) 2.5 / x)## (Intercept) FoodTreatmentSatiated
## Inf 5.007569
## SexParentMale FoodTreatmentSatiated:SexParentMale
## 5.080651 5.679931Assessing priors
Lets now run with priors only so that we can explore the range of values they allow in the posteriors.
owls.rstanarmP1 <- update(owls.rstanP, prior_PD = TRUE)owls.rstanarmP1 %>%
ggpredict(~ FoodTreatment * SexParent) %>%
plot(add.data = TRUE, jitter = c(0.25, 0)) +
scale_y_continuous("", trans = scales::pseudo_log_trans())
Conclusions:
- we see that the range of predictions is fairly wide and the predicted means could range from a small value to a relatively large value.
Defining priors
The following link provides some guidance about defining priors. [https://github.com/stan-dev/stan/wiki/Prior-Choice-Recommendations]
When defining our own priors, we typically do not want them to be scaled.
If we wanted to define our own priors that were less vague, yet still not likely to bias the outcomes, we could try the following priors (mainly plucked out of thin air):
- \(\beta_0\): normal centred at 0 with a standard deviation of 2.5
- \(\beta_1\): normal centred at 0 with a standard deviation of 5
- \(\Sigma\): decov with:
- regularisation: the exponent for a LKJ prior on the correlation matrix. A value of 1 (default) implies a joint uniform prior
- concentration: the concentration parameter for a symmetric Dirichlet distribution. A value of 1 (default) implies a joint uniform distribution
- shape and scale: the shape and scale parameters for a gamma prior on the scale and scale parameters of the decov prior. A value of 1 for both (default) simplifies the gamma prior to a unit-exponential distribution.
I will also overlay the raw data for comparison.
owls.rstanarmP2 <- stan_glmer(NCalls ~ FoodTreatment * SexParent +
offset(log(BroodSize)) + (1 | Nest),
data = owls,
family = poisson(link = "log"),
prior_intercept = normal(0, 2.5, autoscale = FALSE),
prior = normal(0, 5, autoscale = FALSE),
prior_covariance = decov(1, 1, 1, 1),
refresh = 0,
iter = 5000,
prior_PD = TRUE,
warmup = 2000,
thin = 10,
chains = 3,
cores = 3
)owls.rstanarmP2 %>%
ggpredict(~ FoodTreatment * SexParent) %>%
plot(add.data = TRUE, jitter = c(0.25, 0)) +
scale_y_continuous("", trans = scales::pseudo_log_trans())
brms
Using default priors
In brms, the default priors are designed to be weakly informative. They are chosen to provide moderate regularisation (to help prevent over fitting) and help stabilise the computations.
Unlike rstanarm, brms models must be compiled before they start sampling. For most models, the compilation of the stan code takes around 45 seconds.
owls.form <- bf(NCalls ~ FoodTreatment*SexParent +
offset(log(BroodSize)) + (1|Nest),
family=poisson(link='log'))
options(width=150)
owls.form %>% get_prior(data = owls)## prior class coef group resp dpar nlpar bound source
## (flat) b default
## (flat) b FoodTreatmentSatiated (vectorized)
## (flat) b FoodTreatmentSatiated:SexParentMale (vectorized)
## (flat) b SexParentMale (vectorized)
## student_t(3, 1.6, 2.5) Intercept default
## student_t(3, 0, 2.5) sd default
## student_t(3, 0, 2.5) sd Nest (vectorized)
## student_t(3, 0, 2.5) sd Intercept Nest (vectorized)options(width=80)Defining priors
The following link provides some guidance about defining priors. [https://github.com/stan-dev/stan/wiki/Prior-Choice-Recommendations]
When defining our own priors, we typically do not want them to be scaled.
If we wanted to define our own priors that were less vague, yet still not likely to bias the outcomes, we could try the following priors (mainly plucked out of thin air):
- \(\beta_0\): normal centred at 1.8 with a standard deviation of 5
- \(\beta_1\): normal centred at 0 with a standard deviation of 2
- \(\sigma_j\): half-cauchy with parameters 0 and 5.
- \(\Sigma\): decov with:
- regularisation: the exponent for a LKJ prior on the correlation matrix. A value of 1 (default) implies a joint uniform prior
- concentration: the concentration parameter for a symmetric Dirichlet distribution. A value of 1 (default) implies a joint uniform distribution
- shape and scale: the shape and scale parameters for a gamma prior on the scale and scale parameters of the decov prior. A value of 1 for both (default) simplifies the gamma prior to a unit-exponential distribution.
Note, for hierarchical models, the model will tend to want to have a large \(sigma\) in order to fit the data better. It is a good idea to regularise this tendency by applying a prior that has most mass around zero. Suitable candidates include:
- half-t: as the degrees of freedom approach infinity, this will approach a half-normal
- half-cauchy: this is essentially a half-t with 1 degree of freedom
- exponential
I will also overlay the raw data for comparison.
owls %>%
group_by(FoodTreatment, SexParent) %>%
summarise(
log(median(NCalls)),
log(mad(NCalls))
)standist::visualize("normal(1.8,5)", xlim = c(0, 20))
standist::visualize("student_t(3, 0, 2.5)",
"cauchy(0,1)",
xlim = c(-10, 25)
)
priors <- prior(normal(1.8, 5), class = "Intercept") +
prior(normal(0, 2), class = "b") +
prior(cauchy(0, 1), class = "sd")
owls.form <- bf(NCalls ~ FoodTreatment * SexParent +
offset(log(BroodSize)) + (1 | Nest),
family = poisson(link = "log")
)
owls.brm2 <- brm(owls.form,
data = owls,
prior = priors,
sample_prior = "yes",
iter = 5000,
warmup = 2500,
chains = 3,
cores = 3,
thin = 10,
refresh = 0
)
owls.form <- bf(NCalls ~ FoodTreatment * SexParent +
offset(log(BroodSize)) + (FoodTreatment * SexParent | Nest),
family = poisson(link = "log")
)
owls.brm3 <- brm(owls.form,
data = owls,
prior = priors,
sample_prior = "yes",
iter = 5000,
warmup = 2500,
chains = 3,
cores = 3,
thin = 10,
refresh = 0,
control = list(adapt_delta = 0.99)
)
(l.1 <- owls.brm2 %>% loo())##
## Computed from 750 by 599 log-likelihood matrix
##
## Estimate SE
## elpd_loo -2639.3 86.0
## p_loo 148.7 11.7
## looic 5278.7 172.0
## ------
## Monte Carlo SE of elpd_loo is NA.
##
## Pareto k diagnostic values:
## Count Pct. Min. n_eff
## (-Inf, 0.5] (good) 584 97.5% 119
## (0.5, 0.7] (ok) 11 1.8% 60
## (0.7, 1] (bad) 3 0.5% 11
## (1, Inf) (very bad) 1 0.2% 6
## See help('pareto-k-diagnostic') for details.(l.2 <- owls.brm3 %>% loo())##
## Computed from 750 by 599 log-likelihood matrix
##
## Estimate SE
## elpd_loo -2430.8 75.5
## p_loo 301.3 20.8
## looic 4861.6 150.9
## ------
## Monte Carlo SE of elpd_loo is NA.
##
## Pareto k diagnostic values:
## Count Pct. Min. n_eff
## (-Inf, 0.5] (good) 507 84.6% 81
## (0.5, 0.7] (ok) 49 8.2% 51
## (0.7, 1] (bad) 37 6.2% 19
## (1, Inf) (very bad) 6 1.0% 1
## See help('pareto-k-diagnostic') for details.loo_compare(l.1, l.2)## elpd_diff se_diff
## . 0.0 0.0
## . -208.6 47.6owls.brm3 %>% get_variables()## [1] "b_Intercept"
## [2] "b_FoodTreatmentSatiated"
## [3] "b_SexParentMale"
## [4] "b_FoodTreatmentSatiated:SexParentMale"
## [5] "sd_Nest__Intercept"
## [6] "sd_Nest__FoodTreatmentSatiated"
## [7] "sd_Nest__SexParentMale"
## [8] "sd_Nest__FoodTreatmentSatiated:SexParentMale"
## [9] "cor_Nest__Intercept__FoodTreatmentSatiated"
## [10] "cor_Nest__Intercept__SexParentMale"
## [11] "cor_Nest__FoodTreatmentSatiated__SexParentMale"
## [12] "cor_Nest__Intercept__FoodTreatmentSatiated:SexParentMale"
## [13] "cor_Nest__FoodTreatmentSatiated__FoodTreatmentSatiated:SexParentMale"
## [14] "cor_Nest__SexParentMale__FoodTreatmentSatiated:SexParentMale"
## [15] "r_Nest[AutavauxTV,Intercept]"
## [16] "r_Nest[Bochet,Intercept]"
## [17] "r_Nest[Champmartin,Intercept]"
## [18] "r_Nest[ChEsard,Intercept]"
## [19] "r_Nest[Chevroux,Intercept]"
## [20] "r_Nest[CorcellesFavres,Intercept]"
## [21] "r_Nest[Etrabloz,Intercept]"
## [22] "r_Nest[Forel,Intercept]"
## [23] "r_Nest[Franex,Intercept]"
## [24] "r_Nest[GDLV,Intercept]"
## [25] "r_Nest[Gletterens,Intercept]"
## [26] "r_Nest[Henniez,Intercept]"
## [27] "r_Nest[Jeuss,Intercept]"
## [28] "r_Nest[LesPlanches,Intercept]"
## [29] "r_Nest[Lucens,Intercept]"
## [30] "r_Nest[Lully,Intercept]"
## [31] "r_Nest[Marnand,Intercept]"
## [32] "r_Nest[Montet,Intercept]"
## [33] "r_Nest[Murist,Intercept]"
## [34] "r_Nest[Oleyes,Intercept]"
## [35] "r_Nest[Payerne,Intercept]"
## [36] "r_Nest[Rueyes,Intercept]"
## [37] "r_Nest[Seiry,Intercept]"
## [38] "r_Nest[Sevaz,Intercept]"
## [39] "r_Nest[StAubin,Intercept]"
## [40] "r_Nest[Trey,Intercept]"
## [41] "r_Nest[Yvonnand,Intercept]"
## [42] "r_Nest[AutavauxTV,FoodTreatmentSatiated]"
## [43] "r_Nest[Bochet,FoodTreatmentSatiated]"
## [44] "r_Nest[Champmartin,FoodTreatmentSatiated]"
## [45] "r_Nest[ChEsard,FoodTreatmentSatiated]"
## [46] "r_Nest[Chevroux,FoodTreatmentSatiated]"
## [47] "r_Nest[CorcellesFavres,FoodTreatmentSatiated]"
## [48] "r_Nest[Etrabloz,FoodTreatmentSatiated]"
## [49] "r_Nest[Forel,FoodTreatmentSatiated]"
## [50] "r_Nest[Franex,FoodTreatmentSatiated]"
## [51] "r_Nest[GDLV,FoodTreatmentSatiated]"
## [52] "r_Nest[Gletterens,FoodTreatmentSatiated]"
## [53] "r_Nest[Henniez,FoodTreatmentSatiated]"
## [54] "r_Nest[Jeuss,FoodTreatmentSatiated]"
## [55] "r_Nest[LesPlanches,FoodTreatmentSatiated]"
## [56] "r_Nest[Lucens,FoodTreatmentSatiated]"
## [57] "r_Nest[Lully,FoodTreatmentSatiated]"
## [58] "r_Nest[Marnand,FoodTreatmentSatiated]"
## [59] "r_Nest[Montet,FoodTreatmentSatiated]"
## [60] "r_Nest[Murist,FoodTreatmentSatiated]"
## [61] "r_Nest[Oleyes,FoodTreatmentSatiated]"
## [62] "r_Nest[Payerne,FoodTreatmentSatiated]"
## [63] "r_Nest[Rueyes,FoodTreatmentSatiated]"
## [64] "r_Nest[Seiry,FoodTreatmentSatiated]"
## [65] "r_Nest[Sevaz,FoodTreatmentSatiated]"
## [66] "r_Nest[StAubin,FoodTreatmentSatiated]"
## [67] "r_Nest[Trey,FoodTreatmentSatiated]"
## [68] "r_Nest[Yvonnand,FoodTreatmentSatiated]"
## [69] "r_Nest[AutavauxTV,SexParentMale]"
## [70] "r_Nest[Bochet,SexParentMale]"
## [71] "r_Nest[Champmartin,SexParentMale]"
## [72] "r_Nest[ChEsard,SexParentMale]"
## [73] "r_Nest[Chevroux,SexParentMale]"
## [74] "r_Nest[CorcellesFavres,SexParentMale]"
## [75] "r_Nest[Etrabloz,SexParentMale]"
## [76] "r_Nest[Forel,SexParentMale]"
## [77] "r_Nest[Franex,SexParentMale]"
## [78] "r_Nest[GDLV,SexParentMale]"
## [79] "r_Nest[Gletterens,SexParentMale]"
## [80] "r_Nest[Henniez,SexParentMale]"
## [81] "r_Nest[Jeuss,SexParentMale]"
## [82] "r_Nest[LesPlanches,SexParentMale]"
## [83] "r_Nest[Lucens,SexParentMale]"
## [84] "r_Nest[Lully,SexParentMale]"
## [85] "r_Nest[Marnand,SexParentMale]"
## [86] "r_Nest[Montet,SexParentMale]"
## [87] "r_Nest[Murist,SexParentMale]"
## [88] "r_Nest[Oleyes,SexParentMale]"
## [89] "r_Nest[Payerne,SexParentMale]"
## [90] "r_Nest[Rueyes,SexParentMale]"
## [91] "r_Nest[Seiry,SexParentMale]"
## [92] "r_Nest[Sevaz,SexParentMale]"
## [93] "r_Nest[StAubin,SexParentMale]"
## [94] "r_Nest[Trey,SexParentMale]"
## [95] "r_Nest[Yvonnand,SexParentMale]"
## [96] "r_Nest[AutavauxTV,FoodTreatmentSatiated:SexParentMale]"
## [97] "r_Nest[Bochet,FoodTreatmentSatiated:SexParentMale]"
## [98] "r_Nest[Champmartin,FoodTreatmentSatiated:SexParentMale]"
## [99] "r_Nest[ChEsard,FoodTreatmentSatiated:SexParentMale]"
## [100] "r_Nest[Chevroux,FoodTreatmentSatiated:SexParentMale]"
## [101] "r_Nest[CorcellesFavres,FoodTreatmentSatiated:SexParentMale]"
## [102] "r_Nest[Etrabloz,FoodTreatmentSatiated:SexParentMale]"
## [103] "r_Nest[Forel,FoodTreatmentSatiated:SexParentMale]"
## [104] "r_Nest[Franex,FoodTreatmentSatiated:SexParentMale]"
## [105] "r_Nest[GDLV,FoodTreatmentSatiated:SexParentMale]"
## [106] "r_Nest[Gletterens,FoodTreatmentSatiated:SexParentMale]"
## [107] "r_Nest[Henniez,FoodTreatmentSatiated:SexParentMale]"
## [108] "r_Nest[Jeuss,FoodTreatmentSatiated:SexParentMale]"
## [109] "r_Nest[LesPlanches,FoodTreatmentSatiated:SexParentMale]"
## [110] "r_Nest[Lucens,FoodTreatmentSatiated:SexParentMale]"
## [111] "r_Nest[Lully,FoodTreatmentSatiated:SexParentMale]"
## [112] "r_Nest[Marnand,FoodTreatmentSatiated:SexParentMale]"
## [113] "r_Nest[Montet,FoodTreatmentSatiated:SexParentMale]"
## [114] "r_Nest[Murist,FoodTreatmentSatiated:SexParentMale]"
## [115] "r_Nest[Oleyes,FoodTreatmentSatiated:SexParentMale]"
## [116] "r_Nest[Payerne,FoodTreatmentSatiated:SexParentMale]"
## [117] "r_Nest[Rueyes,FoodTreatmentSatiated:SexParentMale]"
## [118] "r_Nest[Seiry,FoodTreatmentSatiated:SexParentMale]"
## [119] "r_Nest[Sevaz,FoodTreatmentSatiated:SexParentMale]"
## [120] "r_Nest[StAubin,FoodTreatmentSatiated:SexParentMale]"
## [121] "r_Nest[Trey,FoodTreatmentSatiated:SexParentMale]"
## [122] "r_Nest[Yvonnand,FoodTreatmentSatiated:SexParentMale]"
## [123] "prior_Intercept"
## [124] "prior_b"
## [125] "prior_sd_Nest"
## [126] "prior_cor_Nest"
## [127] "lp__"
## [128] "accept_stat__"
## [129] "stepsize__"
## [130] "treedepth__"
## [131] "n_leapfrog__"
## [132] "divergent__"
## [133] "energy__"owls.brm3 %>%
hypothesis("FoodTreatmentSatiated=0") %>%
plot()
owls.brm3 %>%
posterior_samples() %>%
dplyr::select(-`lp__`) %>%
pivot_longer(everything(), names_to = "key") %>%
filter(!str_detect(key, "^r")) %>%
mutate(
Type = ifelse(str_detect(key, "prior"), "Prior", "Posterior"),
## Class = ifelse(str_detect(key, 'Intercept'), 'Intercept',
## ifelse(str_detect(key, 'b'), 'b', 'sigma')),
Class = case_when(
str_detect(key, "(^b|^prior).*Intercept$") ~ "Intercept",
str_detect(key, "b_FoodTreatment.*|b_SexParent.*|prior_b") ~ "TREATMENT",
str_detect(key, "sd") ~ "sd",
str_detect(key, "^cor|prior_cor") ~ "cor",
str_detect(key, "sigma") ~ "sigma"
),
Par = str_replace(key, "b_", "")
) %>%
ggplot(aes(x = Type, y = value, color = Par)) +
stat_pointinterval(position = position_dodge(), show.legend = FALSE) +
facet_wrap(~Class, scales = "free")
MCMC sampling diagnostics
In addition to the regular model diagnostics checking, for Bayesian analyses, it is also necessary to explore the MCMC sampling diagnostics to be sure that the chains are well mixed and have converged on a stable posterior.
There are a wide variety of tests that range from the big picture, overall chain characteristics to the very specific detailed tests that allow the experienced modeller to drill down to the very fine details of the chain behaviour. Furthermore, there are a multitude of packages and approaches for exploring these diagnostics.
brms
bayesplot
The bayesplot package offers a range of MCMC diagnostics as well as Posterior Probability Checks (PPC), all of which have a convenient plot() interface. Lets start with the MCMC diagnostics.
available_mcmc()## bayesplot MCMC module:
## mcmc_acf
## mcmc_acf_bar
## mcmc_areas
## mcmc_areas_data
## mcmc_areas_ridges
## mcmc_areas_ridges_data
## mcmc_combo
## mcmc_dens
## mcmc_dens_chains
## mcmc_dens_chains_data
## mcmc_dens_overlay
## mcmc_hex
## mcmc_hist
## mcmc_hist_by_chain
## mcmc_intervals
## mcmc_intervals_data
## mcmc_neff
## mcmc_neff_data
## mcmc_neff_hist
## mcmc_nuts_acceptance
## mcmc_nuts_divergence
## mcmc_nuts_energy
## mcmc_nuts_stepsize
## mcmc_nuts_treedepth
## mcmc_pairs
## mcmc_parcoord
## mcmc_parcoord_data
## mcmc_rank_hist
## mcmc_rank_overlay
## mcmc_recover_hist
## mcmc_recover_intervals
## mcmc_recover_scatter
## mcmc_rhat
## mcmc_rhat_data
## mcmc_rhat_hist
## mcmc_scatter
## mcmc_trace
## mcmc_trace_data
## mcmc_trace_highlight
## mcmc_violinOf these, we will focus on:
- trace: this plots the estimates of each parameter over the post-warmup length of each MCMC chain. Each chain is plotted in a different shade of blue, with each parameter in its own facet. Ideally, each trace should just look like noise without any discernible drift and each of the traces for a specific parameter should look the same (i.e, should not be displaced above or below any other trace for that parameter).
pars <- owls.brm3 %>% get_variables()
pars <- pars %>%
str_extract("^b.Intercept|^b_FoodTreatment.*|^b_SexParent.*|[sS]igma|^sd.*") %>%
na.omit()
pars## [1] "b_Intercept"
## [2] "b_FoodTreatmentSatiated"
## [3] "b_SexParentMale"
## [4] "b_FoodTreatmentSatiated:SexParentMale"
## [5] "sd_Nest__Intercept"
## [6] "sd_Nest__FoodTreatmentSatiated"
## [7] "sd_Nest__SexParentMale"
## [8] "sd_Nest__FoodTreatmentSatiated:SexParentMale"
## attr(,"na.action")
## [1] 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
## [19] 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44
## [37] 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62
## [55] 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80
## [73] 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98
## [91] 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116
## [109] 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133
## attr(,"class")
## [1] "omit"owls.brm3 %>% mcmc_plot(type = "trace", pars = pars)
The chains appear well mixed and very similar
- acf_bar (auto-correlation function): plots the auto-correlation between successive MCMC sample lags for each parameter and each chain
owls.brm3 %>% mcmc_plot(type = "acf_bar", pars = pars)
There is no evidence of auto-correlation in the MCMC samples
- rhat_hist: Rhat is a scale reduction factor measure of convergence between the chains. The closer the values are to 1, the more the chains have converged. Values greater than 1.05 indicate a lack of convergence. There will be an Rhat value for each parameter estimated.
owls.brm3 %>% mcmc_plot(type = "rhat_hist")
All Rhat values are below 1.05, suggesting the chains have converged.
neff_hist (number of effective samples): the ratio of the number of effective samples (those not rejected by the sampler) to the number of samples provides an indication of the effectiveness (and efficiency) of the MCMC sampler. Ratios that are less than 0.5 for a parameter suggest that the sampler spent considerable time in difficult areas of the sampling domain and rejected more than half of the samples (replacing them with the previous effective sample).
If the ratios are low, tightening the priors may help.
owls.brm3 %>% mcmc_plot(type = "neff_hist")
Ratios all very high.
More diagnostics
owls.brm3 %>% mcmc_plot(type = "combo", pars = pars)
owls.brm3 %>% mcmc_plot(type = "violin", pars = pars)
stan plots
The rstan package offers a range of MCMC diagnostics. Lets start with the MCMC diagnostics.
Of these, we will focus on:
- stan_trace: this plots the estimates of each parameter over the post-warmup length of each MCMC chain. Each chain is plotted in a different colour, with each parameter in its own facet. Ideally, each trace should just look like noise without any discernible drift and each of the traces for a specific parameter should look the same (i.e, should not be displaced above or below any other trace for that parameter).
owls.brm3 %>% get_variables()## [1] "b_Intercept"
## [2] "b_FoodTreatmentSatiated"
## [3] "b_SexParentMale"
## [4] "b_FoodTreatmentSatiated:SexParentMale"
## [5] "sd_Nest__Intercept"
## [6] "sd_Nest__FoodTreatmentSatiated"
## [7] "sd_Nest__SexParentMale"
## [8] "sd_Nest__FoodTreatmentSatiated:SexParentMale"
## [9] "cor_Nest__Intercept__FoodTreatmentSatiated"
## [10] "cor_Nest__Intercept__SexParentMale"
## [11] "cor_Nest__FoodTreatmentSatiated__SexParentMale"
## [12] "cor_Nest__Intercept__FoodTreatmentSatiated:SexParentMale"
## [13] "cor_Nest__FoodTreatmentSatiated__FoodTreatmentSatiated:SexParentMale"
## [14] "cor_Nest__SexParentMale__FoodTreatmentSatiated:SexParentMale"
## [15] "r_Nest[AutavauxTV,Intercept]"
## [16] "r_Nest[Bochet,Intercept]"
## [17] "r_Nest[Champmartin,Intercept]"
## [18] "r_Nest[ChEsard,Intercept]"
## [19] "r_Nest[Chevroux,Intercept]"
## [20] "r_Nest[CorcellesFavres,Intercept]"
## [21] "r_Nest[Etrabloz,Intercept]"
## [22] "r_Nest[Forel,Intercept]"
## [23] "r_Nest[Franex,Intercept]"
## [24] "r_Nest[GDLV,Intercept]"
## [25] "r_Nest[Gletterens,Intercept]"
## [26] "r_Nest[Henniez,Intercept]"
## [27] "r_Nest[Jeuss,Intercept]"
## [28] "r_Nest[LesPlanches,Intercept]"
## [29] "r_Nest[Lucens,Intercept]"
## [30] "r_Nest[Lully,Intercept]"
## [31] "r_Nest[Marnand,Intercept]"
## [32] "r_Nest[Montet,Intercept]"
## [33] "r_Nest[Murist,Intercept]"
## [34] "r_Nest[Oleyes,Intercept]"
## [35] "r_Nest[Payerne,Intercept]"
## [36] "r_Nest[Rueyes,Intercept]"
## [37] "r_Nest[Seiry,Intercept]"
## [38] "r_Nest[Sevaz,Intercept]"
## [39] "r_Nest[StAubin,Intercept]"
## [40] "r_Nest[Trey,Intercept]"
## [41] "r_Nest[Yvonnand,Intercept]"
## [42] "r_Nest[AutavauxTV,FoodTreatmentSatiated]"
## [43] "r_Nest[Bochet,FoodTreatmentSatiated]"
## [44] "r_Nest[Champmartin,FoodTreatmentSatiated]"
## [45] "r_Nest[ChEsard,FoodTreatmentSatiated]"
## [46] "r_Nest[Chevroux,FoodTreatmentSatiated]"
## [47] "r_Nest[CorcellesFavres,FoodTreatmentSatiated]"
## [48] "r_Nest[Etrabloz,FoodTreatmentSatiated]"
## [49] "r_Nest[Forel,FoodTreatmentSatiated]"
## [50] "r_Nest[Franex,FoodTreatmentSatiated]"
## [51] "r_Nest[GDLV,FoodTreatmentSatiated]"
## [52] "r_Nest[Gletterens,FoodTreatmentSatiated]"
## [53] "r_Nest[Henniez,FoodTreatmentSatiated]"
## [54] "r_Nest[Jeuss,FoodTreatmentSatiated]"
## [55] "r_Nest[LesPlanches,FoodTreatmentSatiated]"
## [56] "r_Nest[Lucens,FoodTreatmentSatiated]"
## [57] "r_Nest[Lully,FoodTreatmentSatiated]"
## [58] "r_Nest[Marnand,FoodTreatmentSatiated]"
## [59] "r_Nest[Montet,FoodTreatmentSatiated]"
## [60] "r_Nest[Murist,FoodTreatmentSatiated]"
## [61] "r_Nest[Oleyes,FoodTreatmentSatiated]"
## [62] "r_Nest[Payerne,FoodTreatmentSatiated]"
## [63] "r_Nest[Rueyes,FoodTreatmentSatiated]"
## [64] "r_Nest[Seiry,FoodTreatmentSatiated]"
## [65] "r_Nest[Sevaz,FoodTreatmentSatiated]"
## [66] "r_Nest[StAubin,FoodTreatmentSatiated]"
## [67] "r_Nest[Trey,FoodTreatmentSatiated]"
## [68] "r_Nest[Yvonnand,FoodTreatmentSatiated]"
## [69] "r_Nest[AutavauxTV,SexParentMale]"
## [70] "r_Nest[Bochet,SexParentMale]"
## [71] "r_Nest[Champmartin,SexParentMale]"
## [72] "r_Nest[ChEsard,SexParentMale]"
## [73] "r_Nest[Chevroux,SexParentMale]"
## [74] "r_Nest[CorcellesFavres,SexParentMale]"
## [75] "r_Nest[Etrabloz,SexParentMale]"
## [76] "r_Nest[Forel,SexParentMale]"
## [77] "r_Nest[Franex,SexParentMale]"
## [78] "r_Nest[GDLV,SexParentMale]"
## [79] "r_Nest[Gletterens,SexParentMale]"
## [80] "r_Nest[Henniez,SexParentMale]"
## [81] "r_Nest[Jeuss,SexParentMale]"
## [82] "r_Nest[LesPlanches,SexParentMale]"
## [83] "r_Nest[Lucens,SexParentMale]"
## [84] "r_Nest[Lully,SexParentMale]"
## [85] "r_Nest[Marnand,SexParentMale]"
## [86] "r_Nest[Montet,SexParentMale]"
## [87] "r_Nest[Murist,SexParentMale]"
## [88] "r_Nest[Oleyes,SexParentMale]"
## [89] "r_Nest[Payerne,SexParentMale]"
## [90] "r_Nest[Rueyes,SexParentMale]"
## [91] "r_Nest[Seiry,SexParentMale]"
## [92] "r_Nest[Sevaz,SexParentMale]"
## [93] "r_Nest[StAubin,SexParentMale]"
## [94] "r_Nest[Trey,SexParentMale]"
## [95] "r_Nest[Yvonnand,SexParentMale]"
## [96] "r_Nest[AutavauxTV,FoodTreatmentSatiated:SexParentMale]"
## [97] "r_Nest[Bochet,FoodTreatmentSatiated:SexParentMale]"
## [98] "r_Nest[Champmartin,FoodTreatmentSatiated:SexParentMale]"
## [99] "r_Nest[ChEsard,FoodTreatmentSatiated:SexParentMale]"
## [100] "r_Nest[Chevroux,FoodTreatmentSatiated:SexParentMale]"
## [101] "r_Nest[CorcellesFavres,FoodTreatmentSatiated:SexParentMale]"
## [102] "r_Nest[Etrabloz,FoodTreatmentSatiated:SexParentMale]"
## [103] "r_Nest[Forel,FoodTreatmentSatiated:SexParentMale]"
## [104] "r_Nest[Franex,FoodTreatmentSatiated:SexParentMale]"
## [105] "r_Nest[GDLV,FoodTreatmentSatiated:SexParentMale]"
## [106] "r_Nest[Gletterens,FoodTreatmentSatiated:SexParentMale]"
## [107] "r_Nest[Henniez,FoodTreatmentSatiated:SexParentMale]"
## [108] "r_Nest[Jeuss,FoodTreatmentSatiated:SexParentMale]"
## [109] "r_Nest[LesPlanches,FoodTreatmentSatiated:SexParentMale]"
## [110] "r_Nest[Lucens,FoodTreatmentSatiated:SexParentMale]"
## [111] "r_Nest[Lully,FoodTreatmentSatiated:SexParentMale]"
## [112] "r_Nest[Marnand,FoodTreatmentSatiated:SexParentMale]"
## [113] "r_Nest[Montet,FoodTreatmentSatiated:SexParentMale]"
## [114] "r_Nest[Murist,FoodTreatmentSatiated:SexParentMale]"
## [115] "r_Nest[Oleyes,FoodTreatmentSatiated:SexParentMale]"
## [116] "r_Nest[Payerne,FoodTreatmentSatiated:SexParentMale]"
## [117] "r_Nest[Rueyes,FoodTreatmentSatiated:SexParentMale]"
## [118] "r_Nest[Seiry,FoodTreatmentSatiated:SexParentMale]"
## [119] "r_Nest[Sevaz,FoodTreatmentSatiated:SexParentMale]"
## [120] "r_Nest[StAubin,FoodTreatmentSatiated:SexParentMale]"
## [121] "r_Nest[Trey,FoodTreatmentSatiated:SexParentMale]"
## [122] "r_Nest[Yvonnand,FoodTreatmentSatiated:SexParentMale]"
## [123] "prior_Intercept"
## [124] "prior_b"
## [125] "prior_sd_Nest"
## [126] "prior_cor_Nest"
## [127] "lp__"
## [128] "accept_stat__"
## [129] "stepsize__"
## [130] "treedepth__"
## [131] "n_leapfrog__"
## [132] "divergent__"
## [133] "energy__"pars <- owls.brm3 %>% get_variables()
pars <- str_extract(pars, "^b_.*|^sigma$|^sd.*") %>% na.omit()
owls.brm3$fit %>%
stan_trace(pars = pars)
The chains appear well mixed and very similar
- stan_acf (auto-correlation function): plots the auto-correlation between successive MCMC sample lags for each parameter and each chain
owls.brm3$fit %>%
stan_ac(pars = pars)
There is no evidence of auto-correlation in the MCMC samples
- stan_rhat: Rhat is a scale reduction factor measure of convergence between the chains. The closer the values are to 1, the more the chains have converged. Values greater than 1.05 indicate a lack of convergence. There will be an Rhat value for each parameter estimated.
owls.brm3$fit %>% stan_rhat()
All Rhat values are below 1.05, suggesting the chains have converged.
stan_ess (number of effective samples): the ratio of the number of effective samples (those not rejected by the sampler) to the number of samples provides an indication of the effectiveness (and efficiency) of the MCMC sampler. Ratios that are less than 0.5 for a parameter suggest that the sampler spent considerable time in difficult areas of the sampling domain and rejected more than half of the samples (replacing them with the previous effective sample).
If the ratios are low, tightening the priors may help.
owls.brm3$fit %>% stan_ess()
Ratios all very high.
owls.brm3$fit %>%
stan_dens(separate_chains = TRUE, pars = pars)
ggmcmc
The ggmean package also as a set of MCMC diagnostic functions. Lets start with the MCMC diagnostics.
Of these, we will focus on:
- ggs_traceplot: this plots the estimates of each parameter over the post-warmup length of each MCMC chain. Each chain is plotted in a different colour, with each parameter in its own facet. Ideally, each trace should just look like noise without any discernible drift and each of the traces for a specific parameter should look the same (i.e, should not be displaced above or below any other trace for that parameter).
## owls.ggs <- owls.brm3 %>% ggs(burnin = FALSE, inc_warmup = FALSE)
## owls.ggs %>% ggs_traceplot()The chains appear well mixed and very similar
- gss_autocorrelation (autocorrelation function): plots the autocorrelation between successive MCMC sample lags for each parameter and each chain
## ggs_autocorrelation(owls.ggs)There is no evidence of auto-correlation in the MCMC samples
- stan_rhat: Rhat is a scale reduction factor measure of convergence between the chains. The closer the values are to 1, the more the chains have converged. Values greater than 1.05 indicate a lack of convergence. There will be an Rhat value for each parameter estimated.
## ggs_Rhat(owls.ggs)All Rhat values are below 1.05, suggesting the chains have converged.
stan_ess (number of effective samples): the ratio of the number of effective samples (those not rejected by the sampler) to the number of samples provides an indication of the effectiveness (and efficiency) of the MCMC sampler. Ratios that are less than 0.5 for a parameter suggest that the sampler spent considerable time in difficult areas of the sampling domain and rejected more than half of the samples (replacing them with the previous effective sample).
If the ratios are low, tightening the priors may help.
## ggs_effective(owls.ggs)Ratios all very high.
More diagnostics
## ggs_crosscorrelation(owls.ggs)## ggs_grb(owls.ggs)Model validation
brms
pp check
Post predictive checks provide additional diagnostics about the fit of the model. Specifically, they provide a comparison between predictions drawn from the model and the observed data used to train the model.
available_ppc()## bayesplot PPC module:
## ppc_bars
## ppc_bars_grouped
## ppc_boxplot
## ppc_data
## ppc_dens
## ppc_dens_overlay
## ppc_dens_overlay_grouped
## ppc_ecdf_overlay
## ppc_ecdf_overlay_grouped
## ppc_error_binned
## ppc_error_hist
## ppc_error_hist_grouped
## ppc_error_scatter
## ppc_error_scatter_avg
## ppc_error_scatter_avg_vs_x
## ppc_freqpoly
## ppc_freqpoly_grouped
## ppc_hist
## ppc_intervals
## ppc_intervals_data
## ppc_intervals_grouped
## ppc_km_overlay
## ppc_loo_intervals
## ppc_loo_pit
## ppc_loo_pit_data
## ppc_loo_pit_overlay
## ppc_loo_pit_qq
## ppc_loo_ribbon
## ppc_ribbon
## ppc_ribbon_data
## ppc_ribbon_grouped
## ppc_rootogram
## ppc_scatter
## ppc_scatter_avg
## ppc_scatter_avg_grouped
## ppc_stat
## ppc_stat_2d
## ppc_stat_freqpoly_grouped
## ppc_stat_grouped
## ppc_violin_grouped- dens_overlay: plots the density distribution of the observed data (black line) overlayed on top of 50 density distributions generated from draws from the model (light blue). Ideally, the 50 realisations should be roughly consistent with the observed data.
owls.brm3 %>% pp_check(type = "dens_overlay", nsamples = 100)
The model draws appear to differ from the observed data.
- error_scatter_avg: this plots the observed values against the average residuals. Similar to a residual plot, we do not want to see any patterns in this plot. Note, this is not really that useful for models that involve a binomial response
owls.brm3 %>% pp_check(type = "error_scatter_avg")
This is not really interpretable
- intervals: plots the observed data overlayed on top of posterior predictions associated with each level of the predictor. Ideally, the observed data should all fall within the predictive intervals.
owls.brm3 %>% pp_check(group = "Nest", type = "intervals")
The shinystan package allows the full suite of MCMC diagnostics and posterior predictive checks to be accessed via a web interface.
# library(shinystan)
# launch_shinystan(owls.brm2)DHARMa residuals
DHARMa residuals provide very useful diagnostics. Unfortunately, we cannot directly use the simulateResiduals() function to generate the simulated residuals. However, if we are willing to calculate some of the components yourself, we can still obtain the simulated residuals from the fitted stan model.
We need to supply:
- simulated (predicted) responses associated with each observation.
- observed values
- fitted (predicted) responses (averaged) associated with each observation
preds <- owls.brm3 %>% posterior_predict(nsamples = 250, summary = FALSE)
owls.resids <- createDHARMa(
simulatedResponse = t(preds),
observedResponse = owls$NCalls,
fittedPredictedResponse = apply(preds, 2, median),
integerResponse = TRUE
)
plot(owls.resids, quantreg = TRUE)
Conclusions:
- the model does not appear to be a very good fit
- the Q-Q plot deviates substantially from a straight line
- there are outliers
Perhaps we should specifically explore zero-inflation.
owls.resids %>% testZeroInflation()
##
## DHARMa zero-inflation test via comparison to expected zeros with
## simulation under H0 = fitted model
##
## data: simulationOutput
## ratioObsSim = 3.3577, p-value < 2.2e-16
## alternative hypothesis: two.sidedConclusions:
- there is strong evidence of zero-inflation
The data were collected at various times throughout the night. It is possible that this could lead to patterns of dependency that are not already accounted for. For example, perhaps observations that are collected at similar time of the night (within a given nest) have more similar residuals than those at very different time of the night. We can explore whether there are any temporal auto-correlation patterns.
owls.resids %>% testTemporalAutocorrelation(time = owls$ArrivalTime)## Error in testTemporalAutocorrelation(., time = owls$ArrivalTime): testing for temporal autocorrelation requires unique time values - if you have several observations per time value, either use the recalculateResiduals function to aggregate residuals per time step, or extract the residuals from the fitted object, and plot / test each of them independently for temporally repeated subgroups (typical choices would be location / subject etc.). Note that the latter must be done by hand, outside testSpatialAutocorrelation.owls.resid1 <- owls.resids %>% recalculateResiduals(group = interaction(owls$ArrivalTime, owls$Nest), aggregateBy = mean)
owls.resid1 %>% testTemporalAutocorrelation(time = unique(owls$ArrivalTime))
##
## Durbin-Watson test
##
## data: simulationOutput$scaledResiduals ~ 1
## DW = 1.9407, p-value = 0.5966
## alternative hypothesis: true autocorrelation is not 0Conclusions:
- there is no evidence of temporal auto-correlation
Conclusions:
- there is evidence that the model does not fit that well. It is evidently zero inflated and possibly also over-dispersed.
- it would seem that a zero-inflated Poisson or even a zero-inflated Negative Binomial would be a sensible next step.
- zero-inflated models cannot be fit in
glmer(), so we will proceed withglmmTMB()only.
Model refit and validation
Zero-inflated vs hurdle models
- zero-inflated models: are a mixture of Bernoulli and Poisson (or negative binomial) distributions to model situations where it is believed that the observed data are the result of two combined processes (a count process that governs how many items there are to count and a Bernoulli process that governs whether the items are detectable). In this way, zero-inflated models are useful in situations where we believe that some of the zeros are false zeros (recorded as zeros when they should not have been).
- hurdle models: are for situations where the response itself is thought to be the result of two processes: one that governs whether there is a response and then another that governs the value of the response (must be positive).
Fit model 1 (Poisson)
priors <- prior(normal(1.8, 5), class = "Intercept") +
prior(normal(0, 5), class = "b") +
prior(cauchy(0, 1), class = "sd") +
prior(logistic(0, 1), class = "Intercept", dpar = "zi")
owls.form <- bf(NCalls ~ FoodTreatment * SexParent +
offset(log(BroodSize)) +
(FoodTreatment * SexParent | Nest),
zi ~ 1,
family = zero_inflated_poisson(link = "log")
)
owls.brm4 <- brm(owls.form,
data = owls,
prior = priors,
sample_prior = "yes",
iter = 5000,
warmup = 2500,
thin = 10,
chains = 3,
refresh = 0,
cores = 3
)Model validation
preds <- owls.brm4 %>% posterior_predict(nsamples = 250, summary = FALSE)
owls.resids <- createDHARMa(
simulatedResponse = t(preds),
observedResponse = owls$NCalls,
fittedPredictedResponse = apply(preds, 2, median),
integerResponse = TRUE
)
plot(owls.resids, quantreg = TRUE)
owls.resids %>% testZeroInflation()
##
## DHARMa zero-inflation test via comparison to expected zeros with
## simulation under H0 = fitted model
##
## data: simulationOutput
## ratioObsSim = 1.0295, p-value = 0.792
## alternative hypothesis: two.sidedowls.resids %>% testDispersion()
##
## DHARMa nonparametric dispersion test via sd of residuals fitted vs.
## simulated
##
## data: simulationOutput
## ratioObsSim = 1.1987, p-value = 0.048
## alternative hypothesis: two.sidedowls.resids %>% testUniformity()
##
## One-sample Kolmogorov-Smirnov test
##
## data: simulationOutput$scaledResiduals
## D = 0.12482, p-value = 1.569e-08
## alternative hypothesis: two-sidedowls.resids %>% testQuantiles()
##
## Test for location of quantiles via qgam
##
## data: simulationOutput
## p-value = 3.413e-05
## alternative hypothesis: bothowls.resids %>% testResiduals()
## $uniformity
##
## One-sample Kolmogorov-Smirnov test
##
## data: simulationOutput$scaledResiduals
## D = 0.12482, p-value = 1.569e-08
## alternative hypothesis: two-sided
##
##
## $dispersion
##
## DHARMa nonparametric dispersion test via sd of residuals fitted vs.
## simulated
##
## data: simulationOutput
## ratioObsSim = 1.1987, p-value = 0.048
## alternative hypothesis: two.sided
##
##
## $outliers
##
## DHARMa bootstrapped outlier test
##
## data: simulationOutput
## outliers at both margin(s) = 21, observations = 599, p-value < 2.2e-16
## alternative hypothesis: two.sided
## percent confidence interval:
## 0.000000000 0.009223706
## sample estimates:
## outlier frequency (expected: 0.00400667779632721 )
## 0.03505843## $uniformity
##
## One-sample Kolmogorov-Smirnov test
##
## data: simulationOutput$scaledResiduals
## D = 0.12482, p-value = 1.569e-08
## alternative hypothesis: two-sided
##
##
## $dispersion
##
## DHARMa nonparametric dispersion test via sd of residuals fitted vs.
## simulated
##
## data: simulationOutput
## ratioObsSim = 1.1987, p-value = 0.048
## alternative hypothesis: two.sided
##
##
## $outliers
##
## DHARMa bootstrapped outlier test
##
## data: simulationOutput
## outliers at both margin(s) = 21, observations = 599, p-value < 2.2e-16
## alternative hypothesis: two.sided
## percent confidence interval:
## 0.000000000 0.009223706
## sample estimates:
## outlier frequency (expected: 0.00400667779632721 )
## 0.03505843Fit model 2 (Poisson)
priors <- prior(normal(1.8, 5), class = "Intercept") +
prior(normal(0, 5), class = "b") +
prior(cauchy(0, 1), class = "sd") +
prior(logistic(0, 1), class = "Intercept", dpar = "zi") +
prior(normal(0, 1), class = "b", dpar = "zi")
owls.form <- bf(NCalls ~ FoodTreatment * SexParent +
offset(log(BroodSize)) +
(FoodTreatment * SexParent | Nest),
zi ~ FoodTreatment * SexParent,
family = zero_inflated_poisson(link = "log")
)
owls.brm5 <- brm(owls.form,
data = owls,
prior = priors,
sample_prior = "yes",
iter = 5000,
warmup = 2500,
thin = 10,
chains = 3,
refresh = 0,
cores = 3
)Model validation
preds <- owls.brm5 %>% posterior_predict(nsamples = 250, summary = FALSE)
owls.resids <- createDHARMa(
simulatedResponse = t(preds),
observedResponse = owls$NCalls,
fittedPredictedResponse = apply(preds, 2, median),
integerResponse = TRUE
)
plot(owls.resids, quantreg = TRUE)
owls.resids %>% testZeroInflation()
##
## DHARMa zero-inflation test via comparison to expected zeros with
## simulation under H0 = fitted model
##
## data: simulationOutput
## ratioObsSim = 1.0223, p-value = 0.832
## alternative hypothesis: two.sidedowls.resids %>% testDispersion()
##
## DHARMa nonparametric dispersion test via sd of residuals fitted vs.
## simulated
##
## data: simulationOutput
## ratioObsSim = 1.2102, p-value = 0.04
## alternative hypothesis: two.sidedowls.resids %>% testUniformity()
##
## One-sample Kolmogorov-Smirnov test
##
## data: simulationOutput$scaledResiduals
## D = 0.090609, p-value = 0.000107
## alternative hypothesis: two-sidedowls.resids %>% testQuantiles()
##
## Test for location of quantiles via qgam
##
## data: simulationOutput
## p-value = 8.086e-05
## alternative hypothesis: bothowls.resids %>% testResiduals()
## $uniformity
##
## One-sample Kolmogorov-Smirnov test
##
## data: simulationOutput$scaledResiduals
## D = 0.090609, p-value = 0.000107
## alternative hypothesis: two-sided
##
##
## $dispersion
##
## DHARMa nonparametric dispersion test via sd of residuals fitted vs.
## simulated
##
## data: simulationOutput
## ratioObsSim = 1.2102, p-value = 0.04
## alternative hypothesis: two.sided
##
##
## $outliers
##
## DHARMa bootstrapped outlier test
##
## data: simulationOutput
## outliers at both margin(s) = 19, observations = 599, p-value < 2.2e-16
## alternative hypothesis: two.sided
## percent confidence interval:
## 0.00000000 0.01168614
## sample estimates:
## outlier frequency (expected: 0.00452420701168614 )
## 0.03171953## $uniformity
##
## One-sample Kolmogorov-Smirnov test
##
## data: simulationOutput$scaledResiduals
## D = 0.090609, p-value = 0.000107
## alternative hypothesis: two-sided
##
##
## $dispersion
##
## DHARMa nonparametric dispersion test via sd of residuals fitted vs.
## simulated
##
## data: simulationOutput
## ratioObsSim = 1.2102, p-value = 0.04
## alternative hypothesis: two.sided
##
##
## $outliers
##
## DHARMa bootstrapped outlier test
##
## data: simulationOutput
## outliers at both margin(s) = 19, observations = 599, p-value < 2.2e-16
## alternative hypothesis: two.sided
## percent confidence interval:
## 0.00000000 0.01168614
## sample estimates:
## outlier frequency (expected: 0.00452420701168614 )
## 0.03171953Fit model 3 (Negative Binomial)
priors <- prior(normal(1.8, 5), class = "Intercept") +
prior(normal(0, 5), class = "b") +
prior(cauchy(0, 1), class = "sd") +
prior(logistic(0, 1), class = "Intercept", dpar = "zi") +
prior(gamma(0.01, 0.01), class = "shape")
owls.form <- bf(NCalls ~ FoodTreatment * SexParent +
offset(log(BroodSize)) +
(FoodTreatment * SexParent | Nest),
zi ~ 1,
family = zero_inflated_negbinomial(link = "log")
)
owls.brm6 <- brm(owls.form,
data = owls,
prior = priors,
sample_prior = "yes",
iter = 5000,
warmup = 2500,
thin = 10,
chains = 3,
refresh = 0,
cores = 3
)Model validation
preds <- owls.brm6 %>% posterior_predict(nsamples = 250, summary = FALSE)
owls.resids <- createDHARMa(
simulatedResponse = t(preds),
observedResponse = owls$NCalls,
fittedPredictedResponse = apply(preds, 2, median),
integerResponse = TRUE
)
plot(owls.resids, quantreg = TRUE)
owls.resids %>% testZeroInflation()
##
## DHARMa zero-inflation test via comparison to expected zeros with
## simulation under H0 = fitted model
##
## data: simulationOutput
## ratioObsSim = 1.048, p-value = 0.592
## alternative hypothesis: two.sidedowls.resids %>% testDispersion()
##
## DHARMa nonparametric dispersion test via sd of residuals fitted vs.
## simulated
##
## data: simulationOutput
## ratioObsSim = 0.83884, p-value = 0.032
## alternative hypothesis: two.sidedowls.resids %>% testUniformity()
##
## One-sample Kolmogorov-Smirnov test
##
## data: simulationOutput$scaledResiduals
## D = 0.033463, p-value = 0.5136
## alternative hypothesis: two-sidedowls.resids %>% testQuantiles()
##
## Test for location of quantiles via qgam
##
## data: simulationOutput
## p-value = 0.00197
## alternative hypothesis: bothowls.resids %>% testResiduals()
## $uniformity
##
## One-sample Kolmogorov-Smirnov test
##
## data: simulationOutput$scaledResiduals
## D = 0.033463, p-value = 0.5136
## alternative hypothesis: two-sided
##
##
## $dispersion
##
## DHARMa nonparametric dispersion test via sd of residuals fitted vs.
## simulated
##
## data: simulationOutput
## ratioObsSim = 0.83884, p-value = 0.032
## alternative hypothesis: two.sided
##
##
## $outliers
##
## DHARMa bootstrapped outlier test
##
## data: simulationOutput
## outliers at both margin(s) = 0, observations = 599, p-value = 0.18
## alternative hypothesis: two.sided
## percent confidence interval:
## 0.00000000 0.01335559
## sample estimates:
## outlier frequency (expected: 0.00534223706176962 )
## 0## $uniformity
##
## One-sample Kolmogorov-Smirnov test
##
## data: simulationOutput$scaledResiduals
## D = 0.033463, p-value = 0.5136
## alternative hypothesis: two-sided
##
##
## $dispersion
##
## DHARMa nonparametric dispersion test via sd of residuals fitted vs.
## simulated
##
## data: simulationOutput
## ratioObsSim = 0.83884, p-value = 0.032
## alternative hypothesis: two.sided
##
##
## $outliers
##
## DHARMa bootstrapped outlier test
##
## data: simulationOutput
## outliers at both margin(s) = 0, observations = 599, p-value = 0.18
## alternative hypothesis: two.sided
## percent confidence interval:
## 0.00000000 0.01335559
## sample estimates:
## outlier frequency (expected: 0.00534223706176962 )
## 0Fit model 4 (Negative Binomial)
priors <- prior(normal(1.8, 5), class = "Intercept") +
prior(normal(0, 5), class = "b") +
prior(cauchy(0, 1), class = "sd") +
prior(logistic(0, 1), class = "Intercept", dpar = "zi") +
prior(normal(0, 1), class = "b", dpar = "zi") +
prior(gamma(0.01, 0.01), class = "shape")
owls.form <- bf(NCalls ~ FoodTreatment * SexParent +
offset(log(BroodSize)) +
(FoodTreatment * SexParent | Nest),
zi ~ FoodTreatment * SexParent,
family = zero_inflated_negbinomial(link = "log")
)
owls.brm7 <- brm(owls.form,
data = owls,
prior = priors,
sample_prior = "yes",
iter = 5000,
warmup = 2500,
thin = 10,
chains = 3,
refresh = 0,
cores = 3
)Model validation
preds <- owls.brm7 %>% posterior_predict(nsamples = 250, summary = FALSE)
owls.resids <- createDHARMa(
simulatedResponse = t(preds),
observedResponse = owls$NCalls,
fittedPredictedResponse = apply(preds, 2, median),
integerResponse = TRUE
)
plot(owls.resids, quantreg = TRUE)
owls.resids %>% testZeroInflation()
##
## DHARMa zero-inflation test via comparison to expected zeros with
## simulation under H0 = fitted model
##
## data: simulationOutput
## ratioObsSim = 1.0258, p-value = 0.824
## alternative hypothesis: two.sidedowls.resids %>% testDispersion()
##
## DHARMa nonparametric dispersion test via sd of residuals fitted vs.
## simulated
##
## data: simulationOutput
## ratioObsSim = 0.84071, p-value = 0.016
## alternative hypothesis: two.sidedowls.resids %>% testUniformity()
##
## One-sample Kolmogorov-Smirnov test
##
## data: simulationOutput$scaledResiduals
## D = 0.044661, p-value = 0.1832
## alternative hypothesis: two-sidedowls.resids %>% testQuantiles()
##
## Test for location of quantiles via qgam
##
## data: simulationOutput
## p-value = 0.006258
## alternative hypothesis: bothowls.resids %>% testResiduals()
## $uniformity
##
## One-sample Kolmogorov-Smirnov test
##
## data: simulationOutput$scaledResiduals
## D = 0.044661, p-value = 0.1832
## alternative hypothesis: two-sided
##
##
## $dispersion
##
## DHARMa nonparametric dispersion test via sd of residuals fitted vs.
## simulated
##
## data: simulationOutput
## ratioObsSim = 0.84071, p-value = 0.016
## alternative hypothesis: two.sided
##
##
## $outliers
##
## DHARMa bootstrapped outlier test
##
## data: simulationOutput
## outliers at both margin(s) = 0, observations = 599, p-value = 0.08
## alternative hypothesis: two.sided
## percent confidence interval:
## 0.00000000 0.01335559
## sample estimates:
## outlier frequency (expected: 0.00529215358931553 )
## 0## $uniformity
##
## One-sample Kolmogorov-Smirnov test
##
## data: simulationOutput$scaledResiduals
## D = 0.044661, p-value = 0.1832
## alternative hypothesis: two-sided
##
##
## $dispersion
##
## DHARMa nonparametric dispersion test via sd of residuals fitted vs.
## simulated
##
## data: simulationOutput
## ratioObsSim = 0.84071, p-value = 0.016
## alternative hypothesis: two.sided
##
##
## $outliers
##
## DHARMa bootstrapped outlier test
##
## data: simulationOutput
## outliers at both margin(s) = 0, observations = 599, p-value = 0.08
## alternative hypothesis: two.sided
## percent confidence interval:
## 0.00000000 0.01335559
## sample estimates:
## outlier frequency (expected: 0.00529215358931553 )
## 0(l.1 <- loo(owls.brm3))##
## Computed from 750 by 599 log-likelihood matrix
##
## Estimate SE
## elpd_loo -2430.8 75.5
## p_loo 301.3 20.8
## looic 4861.6 150.9
## ------
## Monte Carlo SE of elpd_loo is NA.
##
## Pareto k diagnostic values:
## Count Pct. Min. n_eff
## (-Inf, 0.5] (good) 507 84.6% 81
## (0.5, 0.7] (ok) 49 8.2% 51
## (0.7, 1] (bad) 37 6.2% 19
## (1, Inf) (very bad) 6 1.0% 1
## See help('pareto-k-diagnostic') for details.(l.2 <- loo(owls.brm4))##
## Computed from 750 by 599 log-likelihood matrix
##
## Estimate SE
## elpd_loo -1930.9 53.3
## p_loo 185.3 13.4
## looic 3861.8 106.6
## ------
## Monte Carlo SE of elpd_loo is NA.
##
## Pareto k diagnostic values:
## Count Pct. Min. n_eff
## (-Inf, 0.5] (good) 528 88.1% 88
## (0.5, 0.7] (ok) 42 7.0% 43
## (0.7, 1] (bad) 25 4.2% 19
## (1, Inf) (very bad) 4 0.7% 6
## See help('pareto-k-diagnostic') for details.(l.3 <- loo(owls.brm5))##
## Computed from 750 by 599 log-likelihood matrix
##
## Estimate SE
## elpd_loo -1920.0 53.8
## p_loo 184.4 12.9
## looic 3840.0 107.7
## ------
## Monte Carlo SE of elpd_loo is NA.
##
## Pareto k diagnostic values:
## Count Pct. Min. n_eff
## (-Inf, 0.5] (good) 526 87.8% 62
## (0.5, 0.7] (ok) 48 8.0% 41
## (0.7, 1] (bad) 18 3.0% 15
## (1, Inf) (very bad) 7 1.2% 4
## See help('pareto-k-diagnostic') for details.(l.4 <- loo(owls.brm6))##
## Computed from 750 by 599 log-likelihood matrix
##
## Estimate SE
## elpd_loo -1665.6 28.4
## p_loo 50.7 4.8
## looic 3331.1 56.8
## ------
## Monte Carlo SE of elpd_loo is NA.
##
## Pareto k diagnostic values:
## Count Pct. Min. n_eff
## (-Inf, 0.5] (good) 582 97.2% 150
## (0.5, 0.7] (ok) 13 2.2% 82
## (0.7, 1] (bad) 3 0.5% 23
## (1, Inf) (very bad) 1 0.2% 12
## See help('pareto-k-diagnostic') for details.(l.5 <- loo(owls.brm7))##
## Computed from 750 by 599 log-likelihood matrix
##
## Estimate SE
## elpd_loo -1656.1 29.0
## p_loo 50.7 4.1
## looic 3312.3 57.9
## ------
## Monte Carlo SE of elpd_loo is NA.
##
## Pareto k diagnostic values:
## Count Pct. Min. n_eff
## (-Inf, 0.5] (good) 575 96.0% 167
## (0.5, 0.7] (ok) 17 2.8% 75
## (0.7, 1] (bad) 6 1.0% 64
## (1, Inf) (very bad) 1 0.2% 9
## See help('pareto-k-diagnostic') for details.loo_compare(l.1, l.2, l.3, l.4, l.5)## elpd_diff se_diff
## owls.brm7 0.0 0.0
## owls.brm6 -9.4 5.0
## owls.brm5 -263.8 34.7
## owls.brm4 -274.8 35.0
## owls.brm3 -774.6 66.6Partial effects plots
brms
conditional_effects
owls.brm7 %>%
conditional_effects("FoodTreatment:SexParent") %>%
plot(points = TRUE, jitter = c(0.25, 0))
These predictions appear to be based on the mean BroodSize of approximately 4.39. That is, the predictions are for a nest, not per chick. There does not appear to be a way to indicate the offset value.
ggpredict
owls.brm7 %>%
ggpredict(~ FoodTreatment * SexParent) %>%
plot(add.data = TRUE, jitter = c(0.25, 0))
These predictions appear to be based on the mean BroodSize of approximately 4.39. That is, the predictions are for a nest, not per chick. There does not appear to be a way to indicate the offset value.
ggemmeans
ggemmeans() can accommodate the offset correctly. There are two sensible choices:
- set the offset to the (log) of the mean BroodSize (similar to other partial effects), hence giving predictions appropriate for the average brood size conclusion.
off <- owls %>% summarize(Mean = mean(BroodSize))
as.numeric(off)## [1] 4.392321owls.brm7 %>%
ggemmeans(~ FoodTreatment * SexParent, offset = off$Mean) %>%
plot()
owls.brm7 %>%
ggemmeans(~ FoodTreatment * SexParent, offset = log(off$Mean)) %>%
plot()
owls.brm7 %>%
ggemmeans(~ FoodTreatment * SexParent, offset = log(1)) %>%
plot()
Model investigation
brms
brms captures the MCMC samples from stan within the returned list. There are numerous ways to retrieve and summarise these samples. The first three provide convenient numeric summaries from which you can draw conclusions, the last four provide ways of obtaining the full posteriors.
summary
The summary() method generates simple summaries (mean, standard deviation as well as 10, 50 and 90 percentiles).
owls.brm7 %>% summary()## Family: zero_inflated_negbinomial
## Links: mu = log; shape = identity; zi = logit
## Formula: NCalls ~ FoodTreatment * SexParent + offset(log(BroodSize)) + (FoodTreatment * SexParent | Nest)
## zi ~ FoodTreatment * SexParent
## Data: owls (Number of observations: 599)
## Samples: 3 chains, each with iter = 5000; warmup = 2500; thin = 10;
## total post-warmup samples = 750
##
## Group-Level Effects:
## ~Nest (Number of levels: 27)
## Estimate
## sd(Intercept) 0.30
## sd(FoodTreatmentSatiated) 1.17
## sd(SexParentMale) 0.17
## sd(FoodTreatmentSatiated:SexParentMale) 0.36
## cor(Intercept,FoodTreatmentSatiated) 0.09
## cor(Intercept,SexParentMale) -0.17
## cor(FoodTreatmentSatiated,SexParentMale) -0.23
## cor(Intercept,FoodTreatmentSatiated:SexParentMale) -0.10
## cor(FoodTreatmentSatiated,FoodTreatmentSatiated:SexParentMale) -0.09
## cor(SexParentMale,FoodTreatmentSatiated:SexParentMale) -0.03
## Est.Error
## sd(Intercept) 0.09
## sd(FoodTreatmentSatiated) 0.29
## sd(SexParentMale) 0.12
## sd(FoodTreatmentSatiated:SexParentMale) 0.23
## cor(Intercept,FoodTreatmentSatiated) 0.29
## cor(Intercept,SexParentMale) 0.40
## cor(FoodTreatmentSatiated,SexParentMale) 0.41
## cor(Intercept,FoodTreatmentSatiated:SexParentMale) 0.40
## cor(FoodTreatmentSatiated,FoodTreatmentSatiated:SexParentMale) 0.43
## cor(SexParentMale,FoodTreatmentSatiated:SexParentMale) 0.43
## l-95% CI
## sd(Intercept) 0.14
## sd(FoodTreatmentSatiated) 0.66
## sd(SexParentMale) 0.01
## sd(FoodTreatmentSatiated:SexParentMale) 0.02
## cor(Intercept,FoodTreatmentSatiated) -0.45
## cor(Intercept,SexParentMale) -0.82
## cor(FoodTreatmentSatiated,SexParentMale) -0.86
## cor(Intercept,FoodTreatmentSatiated:SexParentMale) -0.82
## cor(FoodTreatmentSatiated,FoodTreatmentSatiated:SexParentMale) -0.80
## cor(SexParentMale,FoodTreatmentSatiated:SexParentMale) -0.81
## u-95% CI Rhat
## sd(Intercept) 0.50 1.00
## sd(FoodTreatmentSatiated) 1.77 1.00
## sd(SexParentMale) 0.43 1.00
## sd(FoodTreatmentSatiated:SexParentMale) 0.87 1.00
## cor(Intercept,FoodTreatmentSatiated) 0.62 1.00
## cor(Intercept,SexParentMale) 0.71 1.00
## cor(FoodTreatmentSatiated,SexParentMale) 0.69 1.00
## cor(Intercept,FoodTreatmentSatiated:SexParentMale) 0.63 1.00
## cor(FoodTreatmentSatiated,FoodTreatmentSatiated:SexParentMale) 0.75 1.00
## cor(SexParentMale,FoodTreatmentSatiated:SexParentMale) 0.74 1.00
## Bulk_ESS
## sd(Intercept) 762
## sd(FoodTreatmentSatiated) 750
## sd(SexParentMale) 586
## sd(FoodTreatmentSatiated:SexParentMale) 799
## cor(Intercept,FoodTreatmentSatiated) 661
## cor(Intercept,SexParentMale) 754
## cor(FoodTreatmentSatiated,SexParentMale) 853
## cor(Intercept,FoodTreatmentSatiated:SexParentMale) 598
## cor(FoodTreatmentSatiated,FoodTreatmentSatiated:SexParentMale) 929
## cor(SexParentMale,FoodTreatmentSatiated:SexParentMale) 547
## Tail_ESS
## sd(Intercept) 759
## sd(FoodTreatmentSatiated) 753
## sd(SexParentMale) 754
## sd(FoodTreatmentSatiated:SexParentMale) 655
## cor(Intercept,FoodTreatmentSatiated) 695
## cor(Intercept,SexParentMale) 628
## cor(FoodTreatmentSatiated,SexParentMale) 797
## cor(Intercept,FoodTreatmentSatiated:SexParentMale) 670
## cor(FoodTreatmentSatiated,FoodTreatmentSatiated:SexParentMale) 682
## cor(SexParentMale,FoodTreatmentSatiated:SexParentMale) 512
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI
## Intercept 0.84 0.10 0.64 1.06
## zi_Intercept -1.61 0.25 -2.12 -1.15
## FoodTreatmentSatiated -0.77 0.29 -1.37 -0.21
## SexParentMale -0.10 0.11 -0.33 0.10
## FoodTreatmentSatiated:SexParentMale 0.13 0.20 -0.26 0.52
## zi_FoodTreatmentSatiated 0.81 0.34 0.14 1.48
## zi_SexParentMale -0.78 0.34 -1.53 -0.12
## zi_FoodTreatmentSatiated:SexParentMale 0.60 0.44 -0.26 1.52
## Rhat Bulk_ESS Tail_ESS
## Intercept 1.00 810 759
## zi_Intercept 1.00 676 691
## FoodTreatmentSatiated 1.00 634 631
## SexParentMale 1.00 781 682
## FoodTreatmentSatiated:SexParentMale 1.01 662 649
## zi_FoodTreatmentSatiated 1.00 655 630
## zi_SexParentMale 1.00 793 756
## zi_FoodTreatmentSatiated:SexParentMale 1.00 757 799
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## shape 2.75 0.32 2.13 3.34 1.01 812 717
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).Conclusions:
- the average number of calls made per food deprived owl chicks when the female parent returns to the nest is 0.84 (on the link scale). When back-transformed, this is 2.31.
- on average, satiated chicks negotiate -1 (link scale) less when the female returns than deprived chicks. This equates to 0.47 fold fewer times and represents a 53% decline.
- on average, when the male parent returns, deprived chicks negotiate 0 (link scale) less when the female returns. This equates to 0.91 fold fewer times and represents a 9% decline (although this is not significant).
- there was not significantly detectable interaction suggesting that the effect of food treatment was consistent across both parents and vice versa.
- the estimated rate of non-detection of calls (false zeros) for deprived chicks when the female parent returns is -1.61 (logit scale). When back-transformed to the odds scale, this equates to the odds of a zero being false are 0.2:1. Expressed on a probability scale, it would suggest that the probability of a zero being false is 0.17.
- when the chicks are satiated, the odds not detecting a call (false zero) are increased 2.25 fold. That is, the odds increase by 30.76%.
- when the returning parent is male (for deprived chicks), the odds of not detecting a call (false zeros) are reduced 0.46 fold. That is, the odds decline by 68.6%.
- there is substantially more variation in sibling negotiations between food treatment effects within a nest than either between nests or between the sex of the parent effects within nests.
- variation in the sex of the parent effect is slightly negatively correlated to the variation between nests
- variation in the sex of the parent effect is negatively correlated to the variation in the interaction effect.
tidyMCMC
owls.brm7$fit %>%
tidyMCMC(
estimate.method = "median",
conf.int = TRUE, conf.method = "HPDinterval",
rhat = TRUE, ess = TRUE
)Conclusions:
see above
gather_draws
Due to the presence of a log transform in the predictor, it is better to use the regex version.
owls.brm7 %>% get_variables()## [1] "b_Intercept"
## [2] "b_zi_Intercept"
## [3] "b_FoodTreatmentSatiated"
## [4] "b_SexParentMale"
## [5] "b_FoodTreatmentSatiated:SexParentMale"
## [6] "b_zi_FoodTreatmentSatiated"
## [7] "b_zi_SexParentMale"
## [8] "b_zi_FoodTreatmentSatiated:SexParentMale"
## [9] "sd_Nest__Intercept"
## [10] "sd_Nest__FoodTreatmentSatiated"
## [11] "sd_Nest__SexParentMale"
## [12] "sd_Nest__FoodTreatmentSatiated:SexParentMale"
## [13] "cor_Nest__Intercept__FoodTreatmentSatiated"
## [14] "cor_Nest__Intercept__SexParentMale"
## [15] "cor_Nest__FoodTreatmentSatiated__SexParentMale"
## [16] "cor_Nest__Intercept__FoodTreatmentSatiated:SexParentMale"
## [17] "cor_Nest__FoodTreatmentSatiated__FoodTreatmentSatiated:SexParentMale"
## [18] "cor_Nest__SexParentMale__FoodTreatmentSatiated:SexParentMale"
## [19] "shape"
## [20] "r_Nest[AutavauxTV,Intercept]"
## [21] "r_Nest[Bochet,Intercept]"
## [22] "r_Nest[Champmartin,Intercept]"
## [23] "r_Nest[ChEsard,Intercept]"
## [24] "r_Nest[Chevroux,Intercept]"
## [25] "r_Nest[CorcellesFavres,Intercept]"
## [26] "r_Nest[Etrabloz,Intercept]"
## [27] "r_Nest[Forel,Intercept]"
## [28] "r_Nest[Franex,Intercept]"
## [29] "r_Nest[GDLV,Intercept]"
## [30] "r_Nest[Gletterens,Intercept]"
## [31] "r_Nest[Henniez,Intercept]"
## [32] "r_Nest[Jeuss,Intercept]"
## [33] "r_Nest[LesPlanches,Intercept]"
## [34] "r_Nest[Lucens,Intercept]"
## [35] "r_Nest[Lully,Intercept]"
## [36] "r_Nest[Marnand,Intercept]"
## [37] "r_Nest[Montet,Intercept]"
## [38] "r_Nest[Murist,Intercept]"
## [39] "r_Nest[Oleyes,Intercept]"
## [40] "r_Nest[Payerne,Intercept]"
## [41] "r_Nest[Rueyes,Intercept]"
## [42] "r_Nest[Seiry,Intercept]"
## [43] "r_Nest[Sevaz,Intercept]"
## [44] "r_Nest[StAubin,Intercept]"
## [45] "r_Nest[Trey,Intercept]"
## [46] "r_Nest[Yvonnand,Intercept]"
## [47] "r_Nest[AutavauxTV,FoodTreatmentSatiated]"
## [48] "r_Nest[Bochet,FoodTreatmentSatiated]"
## [49] "r_Nest[Champmartin,FoodTreatmentSatiated]"
## [50] "r_Nest[ChEsard,FoodTreatmentSatiated]"
## [51] "r_Nest[Chevroux,FoodTreatmentSatiated]"
## [52] "r_Nest[CorcellesFavres,FoodTreatmentSatiated]"
## [53] "r_Nest[Etrabloz,FoodTreatmentSatiated]"
## [54] "r_Nest[Forel,FoodTreatmentSatiated]"
## [55] "r_Nest[Franex,FoodTreatmentSatiated]"
## [56] "r_Nest[GDLV,FoodTreatmentSatiated]"
## [57] "r_Nest[Gletterens,FoodTreatmentSatiated]"
## [58] "r_Nest[Henniez,FoodTreatmentSatiated]"
## [59] "r_Nest[Jeuss,FoodTreatmentSatiated]"
## [60] "r_Nest[LesPlanches,FoodTreatmentSatiated]"
## [61] "r_Nest[Lucens,FoodTreatmentSatiated]"
## [62] "r_Nest[Lully,FoodTreatmentSatiated]"
## [63] "r_Nest[Marnand,FoodTreatmentSatiated]"
## [64] "r_Nest[Montet,FoodTreatmentSatiated]"
## [65] "r_Nest[Murist,FoodTreatmentSatiated]"
## [66] "r_Nest[Oleyes,FoodTreatmentSatiated]"
## [67] "r_Nest[Payerne,FoodTreatmentSatiated]"
## [68] "r_Nest[Rueyes,FoodTreatmentSatiated]"
## [69] "r_Nest[Seiry,FoodTreatmentSatiated]"
## [70] "r_Nest[Sevaz,FoodTreatmentSatiated]"
## [71] "r_Nest[StAubin,FoodTreatmentSatiated]"
## [72] "r_Nest[Trey,FoodTreatmentSatiated]"
## [73] "r_Nest[Yvonnand,FoodTreatmentSatiated]"
## [74] "r_Nest[AutavauxTV,SexParentMale]"
## [75] "r_Nest[Bochet,SexParentMale]"
## [76] "r_Nest[Champmartin,SexParentMale]"
## [77] "r_Nest[ChEsard,SexParentMale]"
## [78] "r_Nest[Chevroux,SexParentMale]"
## [79] "r_Nest[CorcellesFavres,SexParentMale]"
## [80] "r_Nest[Etrabloz,SexParentMale]"
## [81] "r_Nest[Forel,SexParentMale]"
## [82] "r_Nest[Franex,SexParentMale]"
## [83] "r_Nest[GDLV,SexParentMale]"
## [84] "r_Nest[Gletterens,SexParentMale]"
## [85] "r_Nest[Henniez,SexParentMale]"
## [86] "r_Nest[Jeuss,SexParentMale]"
## [87] "r_Nest[LesPlanches,SexParentMale]"
## [88] "r_Nest[Lucens,SexParentMale]"
## [89] "r_Nest[Lully,SexParentMale]"
## [90] "r_Nest[Marnand,SexParentMale]"
## [91] "r_Nest[Montet,SexParentMale]"
## [92] "r_Nest[Murist,SexParentMale]"
## [93] "r_Nest[Oleyes,SexParentMale]"
## [94] "r_Nest[Payerne,SexParentMale]"
## [95] "r_Nest[Rueyes,SexParentMale]"
## [96] "r_Nest[Seiry,SexParentMale]"
## [97] "r_Nest[Sevaz,SexParentMale]"
## [98] "r_Nest[StAubin,SexParentMale]"
## [99] "r_Nest[Trey,SexParentMale]"
## [100] "r_Nest[Yvonnand,SexParentMale]"
## [101] "r_Nest[AutavauxTV,FoodTreatmentSatiated:SexParentMale]"
## [102] "r_Nest[Bochet,FoodTreatmentSatiated:SexParentMale]"
## [103] "r_Nest[Champmartin,FoodTreatmentSatiated:SexParentMale]"
## [104] "r_Nest[ChEsard,FoodTreatmentSatiated:SexParentMale]"
## [105] "r_Nest[Chevroux,FoodTreatmentSatiated:SexParentMale]"
## [106] "r_Nest[CorcellesFavres,FoodTreatmentSatiated:SexParentMale]"
## [107] "r_Nest[Etrabloz,FoodTreatmentSatiated:SexParentMale]"
## [108] "r_Nest[Forel,FoodTreatmentSatiated:SexParentMale]"
## [109] "r_Nest[Franex,FoodTreatmentSatiated:SexParentMale]"
## [110] "r_Nest[GDLV,FoodTreatmentSatiated:SexParentMale]"
## [111] "r_Nest[Gletterens,FoodTreatmentSatiated:SexParentMale]"
## [112] "r_Nest[Henniez,FoodTreatmentSatiated:SexParentMale]"
## [113] "r_Nest[Jeuss,FoodTreatmentSatiated:SexParentMale]"
## [114] "r_Nest[LesPlanches,FoodTreatmentSatiated:SexParentMale]"
## [115] "r_Nest[Lucens,FoodTreatmentSatiated:SexParentMale]"
## [116] "r_Nest[Lully,FoodTreatmentSatiated:SexParentMale]"
## [117] "r_Nest[Marnand,FoodTreatmentSatiated:SexParentMale]"
## [118] "r_Nest[Montet,FoodTreatmentSatiated:SexParentMale]"
## [119] "r_Nest[Murist,FoodTreatmentSatiated:SexParentMale]"
## [120] "r_Nest[Oleyes,FoodTreatmentSatiated:SexParentMale]"
## [121] "r_Nest[Payerne,FoodTreatmentSatiated:SexParentMale]"
## [122] "r_Nest[Rueyes,FoodTreatmentSatiated:SexParentMale]"
## [123] "r_Nest[Seiry,FoodTreatmentSatiated:SexParentMale]"
## [124] "r_Nest[Sevaz,FoodTreatmentSatiated:SexParentMale]"
## [125] "r_Nest[StAubin,FoodTreatmentSatiated:SexParentMale]"
## [126] "r_Nest[Trey,FoodTreatmentSatiated:SexParentMale]"
## [127] "r_Nest[Yvonnand,FoodTreatmentSatiated:SexParentMale]"
## [128] "prior_Intercept"
## [129] "prior_b"
## [130] "prior_shape"
## [131] "prior_b_zi"
## [132] "prior_Intercept_zi"
## [133] "prior_sd_Nest"
## [134] "prior_cor_Nest"
## [135] "lp__"
## [136] "accept_stat__"
## [137] "stepsize__"
## [138] "treedepth__"
## [139] "n_leapfrog__"
## [140] "divergent__"
## [141] "energy__"owls.draw <- owls.brm7 %>%
gather_draws(`b.Intercept.*|b_FoodTreatment.*|b_SexParent.*`, regex = TRUE)
owls.drawWe can then summarise this
owls.draw %>% median_hdci()owls.brm7 %>%
gather_draws(`b_Intercept.*|b_FoodTreatment.*|b_SexParent.*`, regex = TRUE) %>%
ggplot() +
geom_vline(xintercept = 0, linetype = "dashed") +
stat_slab(aes(
x = .value, y = .variable,
fill = stat(ggdist::cut_cdf_qi(cdf,
.width = c(0.5, 0.8, 0.95),
labels = scales::percent_format()
))
), color = "black") +
scale_fill_brewer("Interval", direction = -1, na.translate = FALSE)
owls.brm7 %>%
gather_draws(`.Intercept.*|b_FoodTreatment.*|b_SexParent.*`, regex = TRUE) %>%
ggplot() +
geom_vline(xintercept = 0, linetype = "dashed") +
stat_halfeye(aes(x = .value, y = .variable)) +
theme_classic()
bayesplot
owls.brm7$fit %>% plot(type = "intervals")
tidy_draws
This is purely a graphical depiction on the posteriors.
owls.brm7 %>% tidy_draws()spread_draws
owls.brm7 %>% spread_draws(`.*Intercept.*|b_FoodTreatment.*|b_SexParent.*`, regex = TRUE)posterior_samples
owls.brm7 %>%
posterior_samples() %>%
as_tibble()\(R^2\)
owls.brm7 %>%
bayes_R2(re.form = NA, summary = FALSE) %>%
median_hdci()owls.brm7 %>%
bayes_R2(re.form = ~ (1 | Nest), summary = FALSE) %>%
median_hdci()owls.brm7 %>%
bayes_R2(re.form = ~ (FoodTreatment * SexParent | Nest), summary = FALSE) %>%
median_hdci()Further investigations
brms
owls.brm7 %>%
emmeans(~FoodTreatment, type = "response")## FoodTreatment prob lower.HPD upper.HPD
## Deprived 2.2 1.865 2.59
## Satiated 1.1 0.516 1.71
##
## Point estimate displayed: median
## Results are back-transformed from the log scale
## HPD interval probability: 0.95## FoodTreatment prob lower.HPD upper.HPD
## Deprived 2.2 1.865 2.59
## Satiated 1.1 0.516 1.71
##
## Point estimate displayed: median
## Results are back-transformed from the log scale
## HPD interval probability: 0.95